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Non-equilibrium quasi-stationary states in a magnetized
plasma
K. Rypdal, J.-V. Paulsen, O. E. Garcia, S. V. Ratynskaia, V. I. Demidov
To cite this version:
K. Rypdal, J.-V. Paulsen, O. E. Garcia, S. V. Ratynskaia, V. I. Demidov. Non-equilibrium
quasi-stationary states in a magnetized plasma. Nonlinear Processes in Geophysics, European Geosciences
Union (EGU), 2003, 10 (1/2), pp.139-149. �hal-00302189�
Nonlinear Processes
in Geophysics
c
European Geosciences Union 2003
Non-equilibrium quasi-stationary states in a magnetized plasma
K. Rypdal, J.-V. Paulsen, O. E. Garcia, S. V. Ratynskaia, and V. I. Demidov
Department of Physics, University of Tromsø, 9037 Tromsø, Norway
Received: 4 January 2002 – Revised: 23 September 2002 – Accepted: 4 October 2002
Abstract. Non-equilibrium quasi-stationary states result-ing from curvature driven interchange instabilities and drift-wave instabilities in a low beta, weakly ionized, magnetized plasma are investigated in the context of laboratory experi-ments in a toroidal configuration. Analytic modelling, nu-merical simulations and experimental results are discussed with emphasis on identifying the unstable modes and under-standing the physics of anomalous particle and energy fluxes and their linkage to self-organized pressure profiles.
1 Introduction
Since K. Birkeland’s pioneering terrella experiments during the first two decades of the 20thcentury (Rypdal and Brundt-land, 1998) laboratory experiments have been carried out to study processes that are assumed to be essential for the global dynamics of the solar-terrestrial system. Birkeland’s experi-ments attempted to model the global system itself, but it has later been shown that it is impossible to scale all relevant physical parameters down to laboratory dimensions (Block, 1967). Many laboratory physicists believe that the scaling problem might invalidate laboratory simulations of global space systems, but insist that there is still a strong case for laboratory investigation of micro-processes. Experiments of this type should be carefully designed to admit quiescent equilibrium states, which then can be tuned to onset of in-stability and turbulence. In this way the linear physics can be fully tested, and linear and non-linear evolution of instabili-ties can be studied and compared to theoretical predictions. Examples of experimental devices that have served this pur-pose are Q-machines (Motley, 1975), where the resistive drift waves were first discovered, and certain large linear devices (Gekelman, 1999), which have facilitated fundamental stud-ies of Alfv´enic phenomena.
Correspondence to: K. Rypdal
(kristoffer.rypdal@phys.uit.no)
One should bear in mind, however, that this experimen-tal focus on the linear dynamics and the transition from sta-ble to unstasta-ble equilibria does not let us study all aspects of the complex dynamics that typically govern space sys-tems, nor is it suitable for studying all relevant instabilities. One example is a class of low-frequency instabilities in mag-netized plasma called interchange instabilities. Such insta-bilities excite field-aligned structures (flute modes) and are driven by a pressure gradient opposing either either a gravity force or a centrifugal force due to magnetic field curvature. They are important in ionospheric dynamics, manifested for instance in the so-called equatorial spread F irregularities, which take their name from the spreading of ionograms ob-tained from equatorial locations (Hysell, 2000). Spread F is now known to be caused by interchange instabilities driven by the steep density gradient in the night-time equatorial F-region. Although only the bottomside F-region is linearly unstable, the nonlinear stage of the instability leads to pen-etration of plumes through the F peak and well into the top side. In this manner the instabilities lead to strong transport and mixing which is of fundamentally nonlinear nature.
Gravity is unimportant in laboratory plasmas, so the study of these phenomena requires a curved magnetic field, while the quiescent laboratory plasmas require a straight field. The problem is that the bending of a simple straight magnetic field not only creates a loss of stability, but also a loss of MHD-equilibrium. In toroidal magnetic confinement devices equilibrium is restored by the introduction of a magnetic ro-tational transform, but this modification also suppresses the interchange instabilities. Thus, neither the quiescent linear devices, nor the toroidal confinement devices are suitable for studying such phenomena.
The philosophy underlying the study of quiescent config-urations for gaining understanding of space systems is based on an implicit assumption that the time-averaged state neces-sarily corresponds to an MHD-equilibrium. However, nat-ural plasmas are often found in quasi-stationary states for which the time-averaged state is not an equilibrium. Such plasmas develop strong stationary or fluctuating flows, and
force balance is maintained only by inclusion of inertial and/or friction forces. The simplest experimental configura-tion for studying such plasma states is the simple magnetized torus (Rypdal et. al., 1994), a configuration where a plasma is immersed in an externally generated toroidal magnetic field without the rotational transform that is necessary for MHD equilibrium and stability.
At this point it may be necessary to make some comments on the notions of equilibrium and stability. Strictly speaking instabilities can only develop on the background of an equi-librium situation, and the notion of instability has no mean-ing if an equilibrium does not exist. Nevertheless, plasma instabilities are often identified in a system with fluctuations, even when the equilibrium state from which the unstable per-turbations grow cannot be uniquely defined. What is identi-fied is the instability mechanism and the fluctuation modes, which may persist also in the nonlinearly saturated state. It will be shown in this paper that a stationary flow can result from the same mechanism that drives the growth of a par-ticular unstable perturbation of an equilibrium state. It is a matter of definition whether such a flow should be con-ceived as a particular saturated state of the instability or a stationary and stable equilibrium flow. We prefer the former definition, because the stationary flow states and the quasi-stationary turbulent states are all governed by the same phys-ical mechanisms. We call a state quasi-stationary if the fluc-tuating density, temperature and fluid velocity are consider-ably smaller than their time-averaged values, and if the fluc-tuations are stationary in the sense that the ensemble mean and auto-correlation are independent of time. In this case it is meaningful to think of the time-averages of these quanti-ties as background profiles.
In a non-equilibrium state the plasma production mecha-nism and the boundary conditions in general force the plasma system to accomodate fluxes of matter, energy and entropy which cannot be maintained in a quiescent state. The classi-cal turbulent transport paradigm could be described as fol-lows: Imposed fluxes universally give rise to gradients in configuration- and/or velocity-space, or to stretching and twisting of magnetic field lines. Such gradients or distor-tions provide the free energy that feed plasma instabilities, and the resulting fluctuations give rise to the transport neces-sary to maintain the imposed fluxes. The linear growth rate of the instability normally increases monotonically with in-creasing gradient, and often there is an instability threshold in the sense that instability occurs only when the gradient exceeds a certain value. For a given imposed flux the dient is given by the negative feedback loop; increased gra-dient → increased growth rate → higher fluctuation levels
→ increased anomalous transport → reduced gradient. If there is no threshold, the relaxed gradient depends strongly on the magnitude of the imposed flux. In the presence of an instability threshold, however, the gradients will usually re-main close to this threshold. This concept of a critical gradi-ent is very useful, although somewhat over-simplified, since the stability of the system may depend on the global pro-file of plasma parameters, not only on local gradients. In
this picture the growth of the unstable modes could in princi-ple saturate before they reach nonlinear levels because of the flattening of the profiles, in analogy with the plateau forma-tion of the particle velocity in quasilinear theory of velocity-space instabilities. The quasilinear picture of turbulence im-plies the existence of a characteristic spatial scale (i.e. the wavelength of the most unstable mode), which appears in the anomalous transport coefficient, and the dynamics of the quasilinear transport is a diffusion process. The quasilin-ear transport paradigm has been challenged by observations from a variety of nonequilibrium quasistationary plasmas in-dicating absence of characteristic scales of the fluctuations. The scale invariance is often manifested in the power-law be-haviour of Fourier energy density spectra of the turbulence. Power-law spectra S(f ) ∼ f−γ at low frequencies have been observed in signals from magnetic confinement plas-mas and space plasplas-mas like those in planetary magnetotails and various astrophysical objects. Such spectra are asso-ciated with an algebraic tail in the autocorrelation function
C(τ ) ∼ τγ −1. If γ > 0, the integralR∞
0 C(τ ) dτ diverges,
and the fluctuations are said to exhibit long-range correla-tions.
The paper is organized as follows. In Sect. 2 we describe the main cross field transport mechanisms for plasma and charge in a low beta and weakly ionized plasma with curved magnetic field, and the physics that determines the plasma potential profile in a hot cathode toroidal discharge is out-lined. Section 3 contains a systematic derivation of a reduced fluid model which contains all the essential physics describ-ing the source driven non-equilibrium quasi-stationary state of such a plasma. A discussion of the (lack of) equilibrium for the simple toroidal plasma configuration and the implica-tions for the resulting plasma state is given in Sect. 4. Sec-tion 5 starts with a summary of known results of local lin-ear stability analysis for flute interchange modes and drift waves, followed by some results of global two-dimensional (2-D) numerical simulations of the model derived in Sect. 3 in the flute mode limit. Stationary and quasi-stationary (tur-bulent) flow states are identified and the existence of criti-cal (resilient) pressure profiles are demonstrated numericriti-cally as well as experimentally. In Sect. 5 we also present flute mode simulations in a slab geometry. Here critical profiles are demonstrated very clearly, and shown to correspond to a linear stability threshold. The plasma potential and elec-tron density fluctuations are shown to be approximately in anti-phase, which is unfavourable for plasma transport. Sec-tion 6 briefly describe some experimental results on fluctu-ations and transport which have been discussed in more de-tailed in a recent publication (Ratynskaia et al., 2002). These results are contrasted with the simulations of Sect. 5, and the role of flute modes and drift waves are discussed.
2 Physics of the simple torus
A valuable supplement to the study of anomalous transport in fusion magnetic confinement devices are experiments on
low frequency instabilities and turbulent transport conducted in cold, magnetized plasma where electrostatic probes can be operated as the main diagnostic. As mentioned in Sect. 1 a configuration that allows excitation of interchange instabili-ties is the simple torus configuration (Rypdal et. al., 1994). This device is simpler than fusion devices in most respects. The magnetic geometry is purely toroidal, simplifying the modelling considerably. Plasma beta is very low, justifying an elecrostatic description of all phenomena. The ions are very cold, allowing us to neglect ion pressure. The electrons are also quite cold, and not very dense, so probes can be used as a diagnostic also in the core plasma. And the device is op-erated in a steady state mode, so device operation represents no limit to the amount of data that can be aqcuired from the experiment.
The plasma in this geometry is most conveniently de-scribed in cylindrical cordinates (R, ϕ, z), where z is the co-ordinate along the major torus axis, R is the distance from this axis, and ϕ is the azimuthal (toroidal) angle. By conven-tion we chose the azimuthal unit vector to be in the opposite direction of the magnetic field, i.e. B = −(B0R0/R) ˆϕ.
2.1 Basic concepts of anomalous transport
In a low beta plasma the time-averaged cross-field anomalous particle flux density 0⊥ = hnv⊥iis due to electrostatic os-cillations in plasma density n and electric field E = −∇⊥φ. The latter gives rise to fluctuations in the plasma drift ve-locity v⊥ ≈vE = ∇⊥φ × ˆϕ/B which follows from Ohm’s law ∇⊥φ = v × B for a perfectly conducting medium. The plasma beta β = µ0p/B2is of order 10−5in these
exper-iments, so the total magnetic field can accurately be repre-sented by the field B = −(B0R0/R) ˆϕ generated by the
ex-ternal field coils. In Sect. 5 it is shown that in this config-uration the electrostatic interchange instability generates ei-ther a stationary flow in the form of a double vortex which transports plasma from the center to the wall by passive ad-vection, or a fluctuating state dominated by poloidally ro-tating, electrostatic field-aligned vortices. In the fluctuat-ing state the vortex dynamics is responsible for a major part the radial mass transport as well as important contributions to the cross-field current. At an arbitrary location in the toroidal plasma column let us consider the flux component along the minor radius, 0x = hnvxi, where vx = Ey/B is the E × B velocity in this direction. Here Ey = −∂yφ is the electric field component in the poloidal direction. By Fourier expanding n and φ in the y-coordinate, we find for the flux density averaged over the poloidal coordinate
0x0 = −(2/B)Pk>0kh|nkφk∗|isin αk(nφ), where αk(nφ) = arg hnkφk∗iis the cross-phase between nk and φk. Essential for the transport is the average phase difference between nk and vk =ikφk/B for the Fourier components of the fields. One observes that the turbulent flux is proportional to the sine of the cross-phase, and hence it vanishes if nkand φkare ex-actly in phase, or exex-actly in antiphase. Cross-field currents can be diamagnetic, inertial, or collisional, as can be shown
by writing the one-fluid momentum equation in the form
j × B = ∇p + ρdv
dt +ρνinv . (1)
Equation (1) shows that the sources of a cross-field current density j are forces due to gradients in the pressure p, inertial forces due to accelerations, and friction forces due to an ion-neutral collision frequency νin. Thus, by taking the cross-product of Eq. (1) by B, the perpendicular current density can be written as j⊥=jp+jm+jν, where jp=B−1∇p ×
ˆ
ϕ, jm=B−1ρ dtv × ˆϕ, and jν =B−1νinρv × ˆϕ.
It will be shown in Sect. 3.1 that for a wide range of plasma parameters we can invoke the quasineutral approximation, which imples that ∇ ·j = 0 even for the instantaneous current density. Thus, an approximate evolution equation is
∇ · jm+jp+jν = −∇ · jq. (2) If v is approximated by the E × B velocity vE, Eq. (2) is an evolution equation for the plasma potential φ (x, t). For axisymmetric perturbations (flute mode turbulence) the term on the right hand side of Eq. (2) vanishes, while for for drift modes it depends on the parallel electron conductivity and the parallel wave number kq. In Sect. 3.4 we present a closed model suitable for numerical simulation, consisting of this evolution equation coupled with the mass continuity equation and an equation of state.
If the plasma is in a state of stationary turbulence, the time-averaged current hjpiis found by the substitution p → hpi in the definition of jp, hjmiby the substitution ρdtv → ∇ ·
hρvvi, and hjνiby the substitution ρv → hρvi. The averaged inertia current hjmican be considered to be driven by the dy-namic stress dyad Tm = hρvvi. In the drift-approximation, where v in Tmis approximated by vE, the inertial current can be recognized as the current arising from the ion polarization drift. The dynamic stress tensor has a contribution from the average flow Tm0 = hρihvihvi and a contribution from the
fluctuations in the flow, the latter is known as the Reynolds stress TR =Tm−Tm0. In a poloidally rotating plasma the
resulting stress (the centrifugal force) gives rise to a poloidal inertia current, and only the Reynolds stress can give rise to a radial component. The current jν driven by the ion-neutral collisions can also be modified by the drift-approximation
v ≈ vE to yield jν ≈ −σP∇⊥φ, where σp = ρνin/B2is known as the Pedersen conductivity. In a two-fluid model this current arises from the perpendicular ion mobility due to ion-neutral collisions.
From Eq. (2) we can draw some important conclusions about the nature of the time-averaged plasma state. In the presence of a stationary, turbulent flow, the charge continuity equation yields ∇ · hj⊥+jqi =0. If the time-averaged state is axisymmetric, we have that ∇ · hjqi =∂ϕhjqi = 0, hence the time-average of Eq. (2) takes the form
2 BR ∂hpi ∂z = ∇ · B × ∇ · Tm B2 −νin∇ · hρ∇⊥φi B2 . (3) For a static situation (v = 0) both terms on the right hand side in Eq. (3) vanish, so equilibrium is possible only if
∂zhpi = 0. This means that a magnetostatic equilibrium with p = 0 on the boundaries cannot exist. An equilibrium in the sense of a stationary, or stationary turbulent, flow can only be provided by the currents hjmiand hjνidue to inertial stress and/or the collisional friction. The charge accumula-tion from these currents is represented by the two terms on the right hand side of Eq. (3). It will be shown in Sect. 4 that global charge- or force-balance can only be obtained if plasma currents are allowed to close along the vessel walls or a poloidal limiter.
2.2 Potential profile and poloidal rotation
The insights presented here have evolved from experiments on plasma discharges in the simple torus “Blaamann” (Ryp-dal et. al., 1994). A standard method of plasma production in a simple torus is by electron emission from a hot negatively biased cathode located near the center line (minor axis) of the toroidal plasma column. This implies that negative charge is injected in a toroidal magnetic flux tube far from the bound-ing walls. The charge accumulatbound-ing in this flux tube can only be compensated by a cross-field electric current flowing in-wards from the wall. Rypdal et al. (1996) have shown that the contribution to the total current from the pressure force and the inertial force from the stationary flow vanishes, hence the discharge current must be anomalous for low neutral gas pressures (< 10−2Pa) (Rypdal et al., 1997), but may be due to ion-neutral collisions for high pressures.
In the experiments a negative potential well is formed with nearly circular equipotentials in a poloidal cross-section, and the potential minimum is located where the flux tube through the cathode intersects this poloidal cross-section plane. The potential well gives rise to a poloidal plasma rotation with a strongly sheared flow in certain regions. This suggests that a radial electric field, and hence a poloidal flow vE = ∇φ ×
ˆ
ϕ/B is neccessary to maintain the cross-field current. In the
straight cylinder limit (R → ∞), and a cylinder-symmetric, laminar poloidal flow, vθ = ∂rφ/B, jp and jmare poloidal and jνis radial. Let us represent the charge injected from the cathode as a source term in the charge continuity equation,
∇ ·jν =S(r), so that I (r) = −2π
Rr
0S(r
0)r0dr0 >0 is the current directed towards the cylinder axis per unit length of the cylinder. The expression for jν can then be integrated to yield φ (r) = − B 2 2π νin Z a r I dr0 ρ(r0)r0, (4) where we have assumed φ (a) = 0 at the wall r = a. If we assume that S is localized to the cylinder axis, i.e. if
S(r) = (2π r)−1δ(r), then I (r) = I0for all r > 0, where I0
is the discharge current per unit length of the cylinder. The potential profile calculated from Eq. (4) agrees very well with the one measured experimentally when electron density mea-surements (needed to estimate ρ(r0)) are made by cylindrical Langmuir probes employing kinetic theory for the electron saturation current in a magnetized plasma (Demidov et al., 1999). This actually serves as a validation of this method
of electron density measurement, which yields values more than 3 times higher than results based on the conventional Langmuir formula for unmagnetized plasmas.
In the real experiment the charge source is also usually a plasma source, since the injected electron also gives rise to ionization. The average number of ionizations per injected electron depends on the discharge voltage. Numerical sim-ulations described in Sect. 5.2 confirm the relaxed potential profiles described by Eq. (4), and they also show that the on-axis plasma density grows linearly due to inefficient radial plasma transport, until the radial density scale-length exceeds some kind of threshold. At this threshold the plasma goes turbulent, and the averaged density gradient remains close to this threshold during the subsequent evolution.
3 Fluid models
The global properties of toroidal plasmas are usually studied within the framework of the MHD-model, while the phenom-ena associated with plasma transport frequently is studied within the electrostatic approximation. In very low beta plas-mas (in the typical simple torus experiments β ∼ 10−5) the electrostatic approximation is always valid. The phenomena we deal with in this paper, plasma equilibrium, flute modes and drift waves, can all be described within the framework of a one-fluid model, although the standard approach is to use a two-fluid model. The former has some appeal, since a formulation in terms of a one-fluid flow and a charge flow (current density) links the electrostatic field directly to the charge continuity equation, and the contributions from ion-ization and charge injection due to electrons emitted from a hot cathode are naturally represented as source terms in these equations.
3.1 The full one-fluid model
By addition of the momentum equations for electrons and ions in a weakly ionized plasma we find a slightly general-ized version of Eq. (1),
ρdv dt = −∇p +j × B + %E − ρνv − ρ me mi νen ∇pe×B enB2 .(5)
Here v = (mivi +meve)/(mi +me) is the one-fluid ve-locity, where ve and vi are the electron and ion fluid veloc-ities, and me and mi the corresponding masses. The quan-tity p = pe +pi = nTe+nTi is the total scalar pressure and % is the charge density. The effect of collisions between electron and ions with neutrals are expressed through the collision frequencies νen and νin, and we have introduced the effective collision frequency ν = νin(1 + n), where
n = meνen/miνin is the ratio between electron and ion mobilities which is of order 10−1for a Helium plasma, and of order 10−2for Argon. In standard one-fluid formulation the remaining information in the momentum equations is ex-pressed in Ohm’s law. For our purpose we express this most conveniently by employing the ion momentum equation to
obtain Ohm’s law perpendicular to the magnetic field. By as-suming ve⊥∼vi⊥, implying vi⊥=v⊥[1 + O(me/mi)], the perpendicular Ohm’s law takes the form
E⊥+v⊥×B = η⊥j + mi e d dt +νin v⊥+ ∇pi en , (6)
and assuming that veq viq, implying veq ≈ −jq/en, we get from the electron momentum equation the parallel Ohm’s law;
Eq=ηjq−
∇qpe
en . (7)
Here η⊥=(me/e2n)νei⊥is the perpendicular Spitzer resis-tivity, and η = (me/e2n)(νei +νen). The remaining fluid equations are the mass continuity equation and the energy equation, ∂ρ ∂t + ∇ ·(ρv) = 0 , (8) d dt pρ −γ = 0 . (9)
We have chosen to represent the energy equation by the adi-abatic equation of state (9), but other closures are of course possible.
We shall invoke the quasi-neutral approximation, which amounts to neglecting the electric force on space charge %E in Eq. (5) and the displacement current in Amp`ere-Maxwell’s law, the latter being equivalent to neglecting ∂%/∂t in the equation ∂%/∂t + ∇ · j = 0. In Goldston and Ruther-ford (1995) it is shown that this approximation is valid if
χ−1 ≡ 0B2/ρ = λ2D/ρs2 1, where χ is the effective electric susceptibility of the plasma, λDis the Debye length, and ρs is the ion Larmor radius with electron temperature. This condition is valid for most magnetized laboratory and space plasmas.
Another approximation we shall make to simplify the dis-cussion is to assume cold ions, so that the ion pressure pican be neglected compared to the electron pressure, and we can put pe≈pin Eqs. (5) and (7). For typical experimental con-ditions, however, the ion pressure term on the right hand side of Eq. (6) is not negligible compared to the other terms on the right hand side, and it will be treated on the same footing as these terms in the following.
3.2 The drift expansion
The drift expansion can be invoked if the terms on the right hand side of Eq. (6) are small compared to each of the terms on the on the left hand side, implying that v⊥ ≈ vE. This is true if the temporal and perpendicular spatial scales of the phenomena in question are given by ω and k, and we have that ω, k⊥v⊥, and νinare all much smaller than the ion cy-clotron frequency ωci =eB/mi. In addition we require that
∇⊥pi/enE⊥ ∼ Ti/eφ and ηj⊥/E⊥ ∼ (Te/eφ)(νei/ωce) are both small compared with unity. For the typical sim-ple torus experiment ω/ωci ∼ νin/ωci ∼ 10−2. The ratios
k⊥v⊥/ωciand Ti/eφcan of course also be determined from the experiments, but there are also some theoretical leads to estimate them. The ion-neutral collisions (charge exhange) (νin ∼ 5 × 104s−1) tend to cool the ions towards the neu-tral gas temperature. However, after a charge exchange col-lision the E × B mechanism accelerates ions to the vE ve-locity within one gyroperiod, and these velocities are ran-domized due to collisions with charged particles. The re-sult is a drifting Maxwellian ion fluid with thermal velocity
vT i ≡ (Ti/mi)1/2 ∼ vE ∼ k⊥φ/B. If we define the fi-nite ion Larmor radius parameter as δi ≡k⊥ρi, this implies that k⊥v⊥/ωci ∼ Ti/eφ ∼ δi. The ratio (Te/eφ)(νei/ωce) is much smaller than δi under all conditions of experimental interest, so perpendicular electron resistivity can safely be neglected. In our laboratory experiments we have δi ∼10−2 for the dominant wavenumbers, hence the leading terms on the right hand side in Eq. (6) are those of order δi. When
δi 1 Eq. (6) can be solved by iteration, which yields
v⊥=vE+v(⊥1)+O(δ
2
ivT i) , (10)
where vE is of order vT i and v(⊥1) = vm+vpi +vν is of order δivT i. Here vm = (Bωci)−1DtE⊥is the inertia drift (ion polarization drift), vpi =(enB2)−1B × ∇pi is the ion diamagnetic drift, and vν =(Bωci)−1νinE⊥is the collisional ion drift giving rise to the cross-field Pedersen current. Note that we have introduced here the notation Dt ≡∂t+vE· ∇. 3.3 The electrostatic model
The electrostatic model assumes that the induced electric field is small compared to the electrostatic component, the ratio of these components being of the order (ω/ kv)β. For typical simple torus parameters ω/ kv ∼ 1 and β ∼ 10−5,
so the electric field is electrostatic. A complete set of equa-tions can the be constructed from Eqs. (5)–(9), and the charge continuity equation by replacing E by −∇φ. By invoking the quasineutral approximation, the term %E in Eq. (5) is neglected, and the charge continuity equation, which fol-lows from the divergence of Ampere’s law, takes the form
∇⊥·j⊥ = −∇qjq. The parallel current jq =j · B/B gives an important contribution to the charge budget if the modes dominating the low frequency fluctuations are drift waves (these have a finite parallel wavenumber so that ∇qjq ∼
kqjq). Since in these cases heat conduction along the field lines may smear out the temperature perturbations, it could be appropriate to replace Eq. (9) with an isothermal equation of state p = c2sρ, where cs2 ≡ Te/mi is the (constant) ion-acoustic velocity. The ES model constitutes a complete set of equations for the fields, v, j, φ, ρ, and p.
3.4 Drift expansion in the electrostatic model
Invoking the drift expansion iteration in Eq. (6) gave us Eq. (10) for the fluid velocity. Replacing v⊥ by the lowest order drift vEin Eq. (5) yields
The inertia driven current jm=(e/mi)ρvmhas been slightly redefined by replacing v⊥by vE, and similarly the Pedersen current jν = (1 + n) (e/mi)ρvνhas been generalized by in-cluding the effect of electron mobility, The last term, j(D)e =
(νen/ωceB)∇p, represents the current due to electron diffu-sion from collidiffu-sions with neutrals, and is of order n com-pared to jν. Normally, therefore, this term is of the same or-der as those of oror-der δi which have already been neglected in the iteration scheme. In order to include the effects of mag-netic field curvature, let us consider a toroidal field in cylin-der coordinates (R, ϕ, z). We assume B = −(B0R0/R) ˆϕ,
and we have ∇ · vE = −(2/RB)(∂φ/∂z), where z is the di-rection along the major torus axis and R is the distance from this axis. Furthermore, by introducing the inverse aspect ra-tio A =a/R0, where a is the minor radius and R0the major
radius of the torus, we obtain
∇ ·(ρv(⊥1)) = − 1 Bωci ∇⊥· [ρ(Dt+νin)∇⊥φ] + O(A, δi), (12) ∇ ·j⊥= − 2 RB ∂p ∂z − 1 B2∇⊥·[ρDt∇⊥φ] + Sn B2+O(A, δi),(13) where Sn = −(1 + n)νin∇⊥·(ρ∇⊥φ) + (νenB/ωce)∇2p represents the charge accumulation from currents driven by collisions with neutrals. From Eq. (7) we have, assuming constant Tealong field lines,
∇qjq= − Te ηe∇ 2 q eφ Te −ln ρ . (14)
Using Eqs. (13), and (14) in the charge continuity equation, we find ∇ ·(ρDt∇⊥φ) = − 2B R ∂p ∂z − TeB2 ηe ∇ 2 q eφ Te −ln ρ +Sn. (15)
Using Eq. (10) the mass continuity equation (8) can be writ-ten as Dtρ = −ρ∇ ·vE− ∇ ·(ρv(⊥1)), which by means of Eqs. (12) and (15) reduces to
Dtρ = 2 RB ρ∂φ ∂z− mi e ∂p ∂z −miTe ηe2 ∇ 2 q eφ Te −ln ρ +Rn, (16) where Rn = (νen/ωceB)(mi/e)∇⊥2p − ∇⊥·(ρ∇⊥φ). In the limit of A → 0 the first terms on the right in Eqs. (15) and (16) vanish, hence to first order in the small parameter
Athese terms represent the effect of toroidicity.
If drift-waves are not excited, the plasma perturbations are “flute-like” and an isothermal assumption is generally not justified. In this case Eqs. (15) and (16) should be com-plemented with an energy equation, describing the evolu-tion of Te. Often it is appropriate to neglect heat conduc-tion in this energy equaconduc-tion, leaving us with the adiabatic equation of state, Eq. (9), which combined with the mass conservation equation (8) takes the form (∂t +v · ∇)p =
−γp∇ · v. By means of Eq. (10) this can be written as
Dtp = −γp∇ ·vE−γp∇ ·v(⊥1)−v (1) ⊥ · ∇p, which reduces to Dtp = 2γp RB ∂φ ∂z + γp Bωci Dt∇⊥2φ + 1 Bωci ∇⊥p · Dt∇⊥φ + Qn, (17)
where Qn =(νin/ωciB)(∇⊥p · ∇⊥φ + γp∇⊥2φ). The drift approximation has allowed us to reduce the numbers of equa-tions and variables from nine in the full electrostatic model to the three Eqs. (15)–(17) in the three scalar variables φ, p, and ρ. This represents a considerable simplification, which is important for the feasibility of numerical simulations as well as for interpretation of simulations and experiments.
For numerical computation Eqs. (15)–(17) have been for-mulated in the cylindrical coordinates. Only axisymmetric perturbations (flute modes) are considered (∂/∂ϕ = 0) so the problem reduces to 2D on a poloidal cross section of the plasma. The resulting equations are solved numerically by means of a finite difference scheme on a square 0.3 m
×0.3 m, and the major radius is R0 =0.6 m (A =0.25). A source term in charge, mass and pressure simulates the effect of emissive cathode located at the center of the cross section. The spatial distribution of the source intensity is a 2-D circularly symmetric Gaussian with standard deviation 0.03 m. Dirichlet boundary contition have been imposed (φ = % = 0, ρ = ρb = const.), and the effect of a cir-cular poloidal limiter near the wall has been modelled as a sink that removes excessive charge and plasma in the limiter region.
Simulations with the geometry described above will be re-ferred to as global. This model describes the experimental situation quite realistically, but since a finite difference code is employed, the code is slow and spatial resolution and accu-racy are rather poor. More accurate and faster codes can be obtained by employing a slab geometry, implementing pe-riodic boundary conditions in the z-direction. This scheme still assumes a plasma source along a vertical (along z) strip in the center of the slab (around R = R0) and a sink (limiter)
at R = R0±a. This model retains the effect of magnetic
field curvature, but charge injection from the source will lead to vertical sheared flow instead of a poloidal rotation. Al-though these local simulations cannot realistically describe the global features of the laboratory experiment, they are more easily linked to analytic theories for the interchange in-stabilitiy. Some results from both classes of simulations will be presented in Sect. 5.
4 The problem of equilibrium
In plasma physics the concept of equilibrium has many dif-ferent meanings, which sometimes leads to some confusion. Full thermodynamic equilibrium is rarely attained because of the long time scales involved in the relaxation of energy between electrons and ions. In magnetically confined plas-mas the time-averaged forces on a fluid element balance each
other, i.e. the plasma exhibits an MHD equilibrium. If the plasma does not attain a stable equilibrium of this type, it is assumed that the fluid is subject to rapid expansion due to the pressure forces and only weak pressure gradients can persist. This picture, however, is too crude to cover all inter-esting laboratory and astrophysical plasmas. Even in the ab-sence of MHD equilibrium magnetic fields can provide con-siderable confinement compared to a field free expansion, al-though the optimal confinement required in magnetic fusion plasmas cannot be obtained. This is the case with the simple torus configuration, where the magnetic field effectively pre-vents expansion along the minor radius, but expansion along the major radius can only be prevented if wall currents are part of the plasma current system.
4.1 Analysis of charge balance
If the wall of the vacuum vessel were electrically insulat-ing, plasma currents in the z direction due to gradient and curvature drift of electrons (assume cold ions) would lead to charge accumulation of opposite sign on the top and bot-tom vessel wall. Note that this current jB = (−2p/BR)ˆz due to guiding center drifts is related to the current density due to fluid drifts through the relation jp =jB+jM, where
jM = ∇ ×(pB/B2)is the magnetization current. The latter, however, does not contribute to charge accumulation on the wall. Note also that jBis equivalent to a current due to grav-itational drift from a centrifugal acceleration gc=2Te/Rmi. The charge accumulation leads to a polarization electric field and to an oppositely directed jm = ρdtE/B2. The sur-face charge density is given by dtσ = jB +jm, and since
E = −σ/0, we get dtE = (ρgc/0B) − (ρ/0)B2dtE, and hence the electric drift along the major radius becomes
vR =
E B =
gc
1 + χ−1t ≈ gct , (18)
where χ−1=0B2/Mnis much smaller than unity for most
laboratory plasmas of interest. Conventional wisdom is that this “free fall” of the plasma in the centrifugal force field makes the simple torus plasma essentially unconfined by the magnetic field. As the reasoning above indicates, however, this is not necessarily true if charge accumulation is pre-vented by a conducting boundary, i.e. if the vertical plasma currrent is closed by a wall current. Such a wall current im-plies that there is a Lorentz force on the wall, and hence (by Newton’s 3. law) that the vacuum vessel can act on the plasma by a force different from the kinetic pressure due to direct wall contact.
4.2 Analysis of force balance
The reasoning above is quite typical for the electrostatic model. The centrifugal force provides a non-solenoidal cur-rent density, and the charge continuity equation requires an additional inertia driven current density which implies a growing electric field and a corresponding accelerated drift velocity. It is also possible to show the same by considering
directly the force balance, and without assuming low beta and the electrostatic approximation. Consider for simplicity a slab like plasma ∂φ =∂z =0. Assume that we have walls at R = R1and R = R2, and let us separate the currents in
the z-direction into plasma currents jpland wall currents jw,
j = jpl+ Iw(1) 2π R1 δ(R − R1) + Iw(2) 2π R2 δ(R − R2) , (19)
where Iw(1,2) are the currents in the inner and outer wall, re-spectively. Assume that the plasma currents and the wall currents constitute a closed circuit (no external circuit), i.e. that Ipl +Iw = 0, where Ipl = 2πR jplR dR and Iw =
Iw(1)+Iw(2). From the R-components of Amp`ere’s law and the momentum equation we get
dp dR =jplB = (jw−j )B = jwB − B µ0R d dR(RB) . (20)
Multiplication of this equation by R2, and integration over the closed interval [R1, R2]including the wall current sheets,
yields 2 Z R2 R1 pR dR =hR2pi R2 R1 +R0B0 2π Iw. (21)
Here we have integrated by parts and used that Ipl +Iw = 0 ⇒ B1R1 = B2R2 = R0B0, where B0 is the magnetic
field generated by the external field coils at R = R0.
Equa-tion (21) shows that the internal plasma pressure can be bal-anced by kinetic pressure due to wall contact or by forces due to wall currents. The latter is of course preferable if one wants to confine the plasma. Let us consider a pressure pro-file that peaks near the center R0=(R1+R2)/2 and is zero at
the left and right boundaries. The diamagnetic current den-sity jp = −B−1dp/dr is in the negative ˆz-direction to the left of the peak, and in the postive direction to the right. Let
Ipl(1) <0 and Ipl(2) >0 be the plasma currents to the left and right of the peak, respectively. If A =a/R0 1, we have
that |Ipl(1)| ≈ |Ipl(2)| ∼ 4π R0hpi/B0, where hpi is the
spa-tially averaged pressure. On the other hand Eq. (21) shows that |Iw| = 4π ahpi/B0, hence Iw ∼ AIpl(1,2). This shows that for a large aspect ratio torus (A 1) the wall current is a small fraction of the current circulating in the plasma.
Even though the global charge/force balance can be tained by wall currents, local balance cannot generally be at-tained in a stationary plasma without flow. Without collisions and flow the total current density is diamagnetic, j⊥ = jp, and we find that ∇ · j⊥ = −(2/BR)∂zp. If the pressure profile is peaked at the center of the plasma, and there is no charge injection, there will be accumulation of charge of op-posite charge at the lower and upper part of the vacuum ves-sel, and hence an E × B-flow along the major radius close to the equatorial plane. Local balance, and hence a static equi-librium, can only be attainded in the special case of a slab configuration, ∂zp =0.
Global simulations of the more general case of a pressure profile peaked at the center also in the z-direction show that
most of the plasma flows back close to the lower and up-per wall, setting up a stationary double vortex flow where the plasma produced in the center is transported to the lim-iter region by advection along the major radius. This shows the existence of stationary flows where the charge accumu-lation due to jp is compensated by the inertial current jm and the Pedersen current jν. The flow equilibria from these simulations agree quite well with our experimental equilib-ria for microwave (ECR) plasma (Rypdal et al., 1997) or for some hot cathode discharges in heavy ion plasma (Paulsen et al., 2000). For more typical hot cathode discharges one has a deep concentric potential well which implies a sheared poloidal rotation of the plasma. As will be discussed in the next section this rotating state is always fluctuating, because only anomalous mechanisms can provide the necessary ra-dial plasma transport.
5 Instabilities and critical gradients
5.1 Instabilities and thresholds
A local stability analysis of flute modes and drift waves in a slab with magnetic field of radius of curvature R0and
ex-ponential density profile n ∼ exp [(R0−R)/Ln]has been performed by Garcia (2001). For a plasma without a sheared flow, flute modes (kq = 0) are unstable only on the outside density slope, where the density gradient points in the oppo-site direction of the radius of curvature of the magnetic field. The instability condition for a mode with vertical wavenum-ber kzand kR=0 is ≡ R0/2Ln>1+ρs2k2z/4, where Lnis the density gradient scale length and ρs =cs/ωciis the Lar-mor radius of an ion with electron temperature. Smaller kz are more unstable than larger, but since the smallest possible wavenumber in a slab with vertical extent Lzis kz=2π/Lz, the instability condition becomes Ln< Lt hr, where
Lt hr = R0 2(1 + π2ρ2 s/L2z) ≈R0 2 . (22)
At the threshold Ln =Lt hrthe smallest possible wavenum-ber is marginally stable and all higher wavenumwavenum-bers are lin-early stable. Stable and marginally stable flute-modes have
αk(nφ)=π.
The threshold for the flute interchange instability is due to the stabilizing effect of the compression term −ρ∇ · vE =
(2ρ/RB)∂zφ on the right hand side of the mass continuity equation (17). This term, and the corresponding threshold, is due to magnetic field curvature and does not occur in the gravitational interchange instability.
While flute modes are driven unstable by field curvature on the weak field side (i.e. on the outside slope) of the torus cross section, drift waves do not require field curvature for instability, and can be unstable both on the outside and inside slopes. However, on the outside slope the compression term creates the same threshold as for flute modes.
t
(ms)
0(V)
0
5
60
3
a a a a Fig. 1. Time evolution of potential minimum φ0for Hydrogen dis-charge (full line) and Argon disdis-charge (broken line).t
(ms)
nmax n t=00
1
3
5
a
a
a
a
Fig. 2. Time evolution of nmax/n0for Hydrogen discharge (full line) and Argon discharge (broken line).
For drift-waves the growth rate and cross phase α(nφ)k de-pend on the parameter
ψ = k2qLnρs ωce νe (1 + k2zρs2)2 k3 zρs3 , (23)
the maximum growth rate ocurring at ψ ≈ 1. In the hy-drodynamic limit ψ 1 one finds that α(nφ)k → π/4,
nk/n0 eφk/Te, and the growth rate γk ∝ k2q/νe. In the adiabatic limit ψ 1 we have αk(nφ)→0, nk/n0≈eφk/Te, and γk ∝νe/ k2q. The smallest possible parallel wavenumber in the toroidal geometry is kq = 1/R0. For the
experimen-tal parameters of the Blaamann device the maximum growth rate (ψ ≈ 1) occurs for kzρz ≈ 1, i.e. for perpendicular wave-lengths λ⊥ ≈ 2 cm. Most of the power is in larger wave-lengths (k2zρs2 1), corresponding to the adiabatic re-gime ψ 1.
The results quoted here are valid for a situation without a shear flow. Studies of the gravitational instability indicate that velocity shear should increase the stability threshold, an some results exist also for curvature driven flute eigenmodes in a slab geometry with prescribed velocity profiles (Mahajan et al., 1997). The relevance of these results to the toroidal geometry is not clear, however.
a a
Fig. 3. Time averaged equidensity contours (left) and equipotential
contours (right) for global simulations of Hydrogen discharge (top) and Argon discharge (bottom). The equipotential contours represent flow surfaces for the E × B-flow and show closed flow surfaces through the source region for the Hydrogen discharge, but open flow surfaces for the Argon discharge, allowing for passive advection of plasma frome source to wall in the latter case.
5.2 Global simulations and experimental profiles
Global simulations of poloidally rotating plasmas in the flute limit (kq = 0) typically show two transient time scales, one short for the formation of the potential structure and one longer for saturation of the density growth. The latter is ba-sically the particle confinement time. During a time of order
νin−1∼0.1 ms after onset of the sources the growth of the po-tential well has saturated. This saturation is shown for a Hy-drogen discharge by the full curve in Fig. 1, and for an Argon discharge with higher plasma density by the broken curve. The potential well for Hydrogen is about 40 V deep, giving rise to flow surfaces which are closed around the source. For Argon the well depth is less than 1 V, giving rise to a situa-tion with open averaged flow surfaces traversing the source region, allowing stationary convective transport. Flow sur-faces for the two cases are shown in Fig. 3.
Figure 2 shows that the density continues to grow for 0.5 ms in the Hydrogen case and for around 5 ms in the Ar-gon case. The slower growth in the ArAr-gon simulation is due to the fact that the initial background electron number den-sity n0was set 10 times higher than for Hydrogen, while the
ionization source was the same. This means that it takes 10 times longer to build up the same density scale length Ln (or equivalently the same ratio, nmax/n0, where nmax is the maximal number density over the cross section of the plasma column). 1 2 100 V 200 V 300 V 400 V p , a.u. e 0 -10 0 10 R - R0, cm
Fig. 4. Measured electron pressure profiles profiles for different
values of the discharge voltage, Vdis =100, 200, 300, 400 V. Ion-ization source is roughly proportional to Vdis.
A technical feature of the simulations shown here is that an initial, spatially homogeneous number density n0 has been
chosen, and the limiter is modelled in such a way that the density in the limiter region never deviates much from this value. In practice this means that n = n0works as a
bound-ary condition. An interesting empirical observation from many simulations is that the time-average hnmaxi/n0seems
to saturate at a value around 2.4 for the turbulent states with closed flow surfaces, and around 2.0 for the quiescent states with stationary flow along open surfaces.
It could be tempting to interpret the universality of the scale length Lnfor turbulent states as the manifestation of a linear stability threshold, and that these states are marginally stable flow equilibria. Marginal stability, however, requires the existence of neighbouring stable states. In the simulation of turbulent states, however, the contribution to ∇ · jmfrom the turbulent stress is a significant contribution to the charge balance, and this indicates that the time-averaged concentric flow states are not equilibria, and thus cannot correspond to marginal stability. Thus, it is conceivable that the univer-sal value of Ln is a characteristic property of the nonlinear saturated states in this geometric configuration rather than a manifestation of a linear stability threshold.
Experimental electron pressure profiles for different mag-nitudes of the plasma source (discharge voltage) are shown in Fig. 4. The radial profile on the outside slope seems to be close to exponential, and the scale length is nearly in-dependent of the source strength. The pressure gradient is almost ten times steeper than that corresponding to the lin-ear instability threshold in a plasma without shlin-ear flow given by Eq. (22), but agrees well with the gradient scale length observed in the global simulations.
R hni n t=0 1:0 1:5 a a a a
Fig. 5. Electron density profiles from local simulations without
charge source for plasma source of strength 1, 10, and 100, respec-tively.
5.3 Local simulations
Criticality of gradients are observed more clearly in local simulations in slab geometry, where sources and limiters are vertical strips (in z-direction), and we have periodic bound-ary conditions in this direction. This provides a faster code, but cannot provide poloidal rotation of the plasma. On the other hand, this geometry provides the existence of an equi-librium, which gives clearer meaning to the concept of sta-bility.
Without charge source the density gradient on the outside slope is very close to (slightly above) the one correspond-ing to the linear stability threshold, and this feature is very insensitive to the source strength. Radial profiles for differ-ent source strengths (but without sheared flows) are shown in Fig. 5. The profiles on the outside slope are nearly expo-nential and Lnis almost independent of the source and very close to the linear threshold Lt hr =R0/2.
Spatial spectral analysis of the turbulent particle flux shows that the largest scales (smallest wavenumbers) are re-sponsible for most of the total flux. The cross phase αk(nφ)is somewhat above π (α(nφ)k =π yields zero anomalous flux), and power spectra are scale invariant (have power-law depen-dence ∼ f−α). Slab simulations with charge source yields a potential well and sheared flow in the z-direction. Like the non-slab simulations the saturated gradient now becomes much steeper than the linear threshold for unsheared flow. The velocity shear brings the cross-phase α(nφ)k very close to π , i.e. very unfavourable for transport, hence requiring stronger fluctuations to sustain a given flux. With a realistic ion-neutral collision frequency introduced in the model, no self-generated zonal flows appear in these simulations. How-ever, if the realistic dissipation terms are replaced with only a diffusion term in the mass continuity equation, zonal sheared flows develop in the absence of a charge source. These flows effectively quench the instability until a stationary, nonturbu-lent state appears with a purely diffusive density profile.
6 Experimental results on fluctuations and transport
A challenging diagnostic problem in the context of cross-field anomalous transport is to perform accurate local mea-surements of instantaneous particle and energy density fluxes. The basic problem is the following: Suppose the z-axis of a cartesian coordinate system is directed along the ambient magnetic field and we want to measure the anoma-lous flux in the the x-direction. Then simultaneous measure-ments of instantaneous values of electron density n, electric field Ey and electron temperature Te are necessary to ob-tain the anomalous cross-field flux densities. If the dominant modes have cross-phases αnφ(ω)close to 0 or π , the respec-tive fluxes (which are proportional to sin αnφ) become very sensitive to errors in the measurements of αnφ. Such errors will arise for instance if one does not succeed in eliminat-ing the influence of ˜Te on the measurements of ˜Vp and ˜n. A method to deal with this problem in the context of Lang-muir probe measurements was developed by Ratynskaia et al. (2000a,b), and measurements of fluctuations, cross-phases and anomalous fluxes for the simple torus configuration have recently been published by Ratynskaia et al. (2002). The conclusion of these measurements are that flute modes dom-inate all fluctuations on the outer density slope (on the weak field side), while drift waves might dominate density and po-tential fluctuations on the inner slope, where flute modes are locally stable. Weak flute modes coexist with the drift waves on the inside slope, manifested through weak temperature fluctuations. The flute mode spectrum exhibits a peak which is due to large scale poloidally rotating structures, probably associated with the lowest poloidal wavenumber (m = 1). This is the fastest growing mode according to linear theory for flute interchange instability, but if the gradient is at the threshold for this instability, it also is the only linearly unsta-ble mode. If this is the case, the higher wave-numbers (cor-responding to the power-law spectrum at higher frequencies) are nonlinearly driven trough a turbulent cascade. The flux measurements show that the large scale flute mode structures play a substantial role in both particle and energy transport on the outside. On the inside only drift waves contribute to particle transport, while the coupling between temperature fluctuations in flute modes and electric field fluctuations in drift waves yields the major contribution to the energy trans-port.
7 Conclusions
The plasma parameters of many small scale magnetized lab-oratory devices require mathematical models for description of the low frequency dynamics which are remarkably simi-lar to those employed for description for ionospheric inter-change turbulence and field-aligned irregularities. Our elec-trostatic model derived in Sect. 3.4 (Eqs. 15–17) are general-izations of the model for ionospheric turbulence derived by Huba et al. (1985) and applied to this problem by Hassam et al. (1985). On the other hand this model is also a
gener-alization of generic models for drift wave turbulence, which has been applied to ionospheric problems as well as labora-tory plasmas, including edge turbulence in magnetic confine-ment devices.
The numerical and experimental studies of plasma poten-tial profiles and potenpoten-tial fluctuations show the existence of both quiet stationary flow states with slow flow along open flow surfaces, and of turbulent flow states with a fast mean poloidal flow. The transition between these types of states has so far not been studied systematically. Time-averaged density (or pressure) profiles indicate the existence of criti-cal profiles in the sense that the gradient scriti-cale lengths attain characteristic values that are resilient to variation of the im-posed fluxes or other discharge parameters. The absence of a static or stationary flow equilibrium state close to the time-averaged states observed in global simulations and experi-ments indicates that this resiliency is not simply a manifesta-tion of a linear stability threshold. Further study is required before the true nature of the critical profiles is properly un-derstood.
Experimental identification of the turbulent modes and measurement of anomalous particle and energy flux densities were briefly reported in Sect. 6. The modes on the weak field side (outside) were identified as electrostatic flute modes, and hence driven by the interchange instability. On the strong field side these modes are stable, and measurements of cross-phases between density and electric field fluctuations, and of dispersion characteristics (phase velocity), indicate that drift waves dominate the density and electric field fluctuations in this region. The possible role of drift waves indicates that parallel electron dynamics should be included in the numeri-cal models, and that 3-D simulation may reveal new physics. Flux measurements show that both the large scale coher-ent structures and the power-law part of the turbulcoher-ent wave-number spectrum contribute to the total anomalous fluxes. Flute modes and drift waves may both play a role in the trans-port.
Acknowledgements. This work was supported by the Research
Council of Norway under grants 125713/410 and 135373/432.
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